/* specfunc/ellint.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Author: G. Jungman */
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_precision.h>
#include <gsl/gsl_sf_ellint.h>
#include "error.h"
/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
inline
static double locMAX3(double x, double y, double z)
{
double xy = GSL_MAX(x, y);
return GSL_MAX(xy, z);
}
inline
static double locMAX4(double x, double y, double z, double w)
{
double xy = GSL_MAX(x, y);
double xyz = GSL_MAX(xy, z);
return GSL_MAX(xyz, w);
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
/* based on Carlson's algorithms:
[B. C. Carlson Numer. Math. 33, 1 (1979)]
see also:
[B.C. Carlson, Special Functions of Applied Mathematics (1977)]
*/
/* According to Carlson's algorithm, the errtol parameter
typically effects the relative error in the following way:
relative error < 16 errtol^6 / (1 - 2 errtol)
errtol precision
------ ----------
0.001 1.0e-17
0.003 2.0e-14
0.01 2.0e-11
0.03 2.0e-8
0.1 2.0e-5
*/
int
gsl_sf_ellint_RC_e(double x, double y, gsl_mode_t mode, gsl_sf_result * result)
{
const double lolim = 5.0 * GSL_DBL_MIN;
const double uplim = 0.2 * GSL_DBL_MAX;
const gsl_prec_t goal = GSL_MODE_PREC(mode);
const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
const double prec = gsl_prec_eps[goal];
const int nmax = 10000;
if(x < 0.0 || y < 0.0 || x + y < lolim) {
DOMAIN_ERROR(result);
}
else if(GSL_MAX(x, y) < uplim) {
const double c1 = 1.0 / 7.0;
const double c2 = 9.0 / 22.0;
double xn = x;
double yn = y;
double mu, sn, lamda, s;
int n = 0;
while(1) {
mu = (xn + yn + yn) / 3.0;
sn = (yn + mu) / mu - 2.0;
if (fabs(sn) < errtol) break;
lamda = 2.0 * sqrt(xn) * sqrt(yn) + yn;
xn = (xn + lamda) * 0.25;
yn = (yn + lamda) * 0.25;
n++;
if(n==nmax) {
MAXITER_ERROR (result);
}
}
s = sn * sn * (0.3 + sn * (c1 + sn * (0.375 + sn * c2)));
result->val = (1.0 + s) / sqrt(mu);
result->err = prec * fabs(result->val);
return GSL_SUCCESS;
}
else {
DOMAIN_ERROR(result);
}
}
int
gsl_sf_ellint_RD_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
{
const gsl_prec_t goal = GSL_MODE_PREC(mode);
const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
const double prec = gsl_prec_eps[goal];
const double lolim = 2.0/pow(GSL_DBL_MAX, 2.0/3.0);
const double uplim = pow(0.1*errtol/GSL_DBL_MIN, 2.0/3.0);
const int nmax = 10000;
if(GSL_MIN(x,y) < 0.0 || GSL_MIN(x+y,z) < lolim) {
DOMAIN_ERROR(result);
}
else if(locMAX3(x,y,z) < uplim) {
const double c1 = 3.0 / 14.0;
const double c2 = 1.0 / 6.0;
const double c3 = 9.0 / 22.0;
const double c4 = 3.0 / 26.0;
double xn = x;
double yn = y;
double zn = z;
double sigma = 0.0;
double power4 = 1.0;
double ea, eb, ec, ed, ef, s1, s2;
double mu, xndev, yndev, zndev;
int n = 0;
while(1) {
double xnroot, ynroot, znroot, lamda;
double epslon;
mu = (xn + yn + 3.0 * zn) * 0.2;
xndev = (mu - xn) / mu;
yndev = (mu - yn) / mu;
zndev = (mu - zn) / mu;
epslon = locMAX3(fabs(xndev), fabs(yndev), fabs(zndev));
if (epslon < errtol) break;
xnroot = sqrt(xn);
ynroot = sqrt(yn);
znroot = sqrt(zn);
lamda = xnroot * (ynroot + znroot) + ynroot * znroot;
sigma += power4 / (znroot * (zn + lamda));
power4 *= 0.25;
xn = (xn + lamda) * 0.25;
yn = (yn + lamda) * 0.25;
zn = (zn + lamda) * 0.25;
n++;
if(n==nmax) {
MAXITER_ERROR (result);
}
}
ea = xndev * yndev;
eb = zndev * zndev;
ec = ea - eb;
ed = ea - 6.0 * eb;
ef = ed + ec + ec;
s1 = ed * (- c1 + 0.25 * c3 * ed - 1.5 * c4 * zndev * ef);
s2 = zndev * (c2 * ef + zndev * (- c3 * ec + zndev * c4 * ea));
result->val = 3.0 * sigma + power4 * (1.0 + s1 + s2) / (mu * sqrt(mu));
result->err = prec * fabs(result->val);
return GSL_SUCCESS;
}
else {
DOMAIN_ERROR(result);
}
}
int
gsl_sf_ellint_RF_e(double x, double y, double z, gsl_mode_t mode, gsl_sf_result * result)
{
const double lolim = 5.0 * GSL_DBL_MIN;
const double uplim = 0.2 * GSL_DBL_MAX;
const gsl_prec_t goal = GSL_MODE_PREC(mode);
const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
const double prec = gsl_prec_eps[goal];
const int nmax = 10000;
if(x < 0.0 || y < 0.0 || z < 0.0) {
DOMAIN_ERROR(result);
}
else if(x+y < lolim || x+z < lolim || y+z < lolim) {
DOMAIN_ERROR(result);
}
else if(locMAX3(x,y,z) < uplim) {
const double c1 = 1.0 / 24.0;
const double c2 = 3.0 / 44.0;
const double c3 = 1.0 / 14.0;
double xn = x;
double yn = y;
double zn = z;
double mu, xndev, yndev, zndev, e2, e3, s;
int n = 0;
while(1) {
double epslon, lamda;
double xnroot, ynroot, znroot;
mu = (xn + yn + zn) / 3.0;
xndev = 2.0 - (mu + xn) / mu;
yndev = 2.0 - (mu + yn) / mu;
zndev = 2.0 - (mu + zn) / mu;
epslon = locMAX3(fabs(xndev), fabs(yndev), fabs(zndev));
if (epslon < errtol) break;
xnroot = sqrt(xn);
ynroot = sqrt(yn);
znroot = sqrt(zn);
lamda = xnroot * (ynroot + znroot) + ynroot * znroot;
xn = (xn + lamda) * 0.25;
yn = (yn + lamda) * 0.25;
zn = (zn + lamda) * 0.25;
n++;
if(n==nmax) {
MAXITER_ERROR (result);
}
}
e2 = xndev * yndev - zndev * zndev;
e3 = xndev * yndev * zndev;
s = 1.0 + (c1 * e2 - 0.1 - c2 * e3) * e2 + c3 * e3;
result->val = s / sqrt(mu);
result->err = prec * fabs(result->val);
return GSL_SUCCESS;
}
else {
DOMAIN_ERROR(result);
}
}
int
gsl_sf_ellint_RJ_e(double x, double y, double z, double p, gsl_mode_t mode, gsl_sf_result * result)
{
const gsl_prec_t goal = GSL_MODE_PREC(mode);
const double errtol = ( goal == GSL_PREC_DOUBLE ? 0.001 : 0.03 );
const double prec = gsl_prec_eps[goal];
const double lolim = pow(5.0 * GSL_DBL_MIN, 1.0/3.0);
const double uplim = 0.3 * pow(0.2 * GSL_DBL_MAX, 1.0/3.0);
const int nmax = 10000;
if(x < 0.0 || y < 0.0 || z < 0.0) {
DOMAIN_ERROR(result);
}
else if(x + y < lolim || x + z < lolim || y + z < lolim || p < lolim) {
DOMAIN_ERROR(result);
}
else if(locMAX4(x,y,z,p) < uplim) {
const double c1 = 3.0 / 14.0;
const double c2 = 1.0 / 3.0;
const double c3 = 3.0 / 22.0;
const double c4 = 3.0 / 26.0;
double xn = x;
double yn = y;
double zn = z;
double pn = p;
double sigma = 0.0;
double power4 = 1.0;
double mu, xndev, yndev, zndev, pndev;
double ea, eb, ec, e2, e3, s1, s2, s3;
int n = 0;
while(1) {
double xnroot, ynroot, znroot;
double lamda, alfa, beta;
double epslon;
gsl_sf_result rcresult;
int rcstatus;
mu = (xn + yn + zn + pn + pn) * 0.2;
xndev = (mu - xn) / mu;
yndev = (mu - yn) / mu;
zndev = (mu - zn) / mu;
pndev = (mu - pn) / mu;
epslon = locMAX4(fabs(xndev), fabs(yndev), fabs(zndev), fabs(pndev));
if(epslon < errtol) break;
xnroot = sqrt(xn);
ynroot = sqrt(yn);
znroot = sqrt(zn);
lamda = xnroot * (ynroot + znroot) + ynroot * znroot;
alfa = pn * (xnroot + ynroot + znroot) + xnroot * ynroot * znroot;
alfa = alfa * alfa;
beta = pn * (pn + lamda) * (pn + lamda);
rcstatus = gsl_sf_ellint_RC_e(alfa, beta, mode, &rcresult);
if(rcstatus != GSL_SUCCESS) {
result->val = 0.0;
result->err = 0.0;
return rcstatus;
}
sigma += power4 * rcresult.val;
power4 *= 0.25;
xn = (xn + lamda) * 0.25;
yn = (yn + lamda) * 0.25;
zn = (zn + lamda) * 0.25;
pn = (pn + lamda) * 0.25;
n++;
if(n==nmax) {
MAXITER_ERROR (result);
}
}
ea = xndev * (yndev + zndev) + yndev * zndev;
eb = xndev * yndev * zndev;
ec = pndev * pndev;
e2 = ea - 3.0 * ec;
e3 = eb + 2.0 * pndev * (ea - ec);
s1 = 1.0 + e2 * (- c1 + 0.75 * c3 * e2 - 1.5 * c4 * e3);
s2 = eb * (0.5 * c2 + pndev * (- c3 - c3 + pndev * c4));
s3 = pndev * ea * (c2 - pndev * c3) - c2 * pndev * ec;
result->val = 3.0 * sigma + power4 * (s1 + s2 + s3) / (mu * sqrt(mu));
result->err = prec * fabs(result->val);
return GSL_SUCCESS;
}
else {
DOMAIN_ERROR(result);
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.1)] */
int
gsl_sf_ellint_F_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
{
/* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
exact reduction but this will have to do for now) BJG */
double nc = floor(phi/M_PI + 0.5);
double phi_red = phi - nc * M_PI;
phi = phi_red;
{
double sin_phi = sin(phi);
double sin2_phi = sin_phi*sin_phi;
double x = 1.0 - sin2_phi;
double y = 1.0 - k*k*sin2_phi;
gsl_sf_result rf;
int status = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
result->val = sin_phi * rf.val;
result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(sin_phi*rf.err);
if (nc == 0) {
return status;
} else {
gsl_sf_result rk; /* add extra terms from periodicity */
const int rkstatus = gsl_sf_ellint_Kcomp_e(k, mode, &rk);
result->val += 2*nc*rk.val;
result->err += 2*fabs(nc)*rk.err;
return GSL_ERROR_SELECT_2(status, rkstatus);
}
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.2)] */
int
gsl_sf_ellint_E_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
{
/* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
exact reduction but this will have to do for now) BJG */
double nc = floor(phi/M_PI + 0.5);
double phi_red = phi - nc * M_PI;
phi = phi_red;
{
const double sin_phi = sin(phi);
const double sin2_phi = sin_phi * sin_phi;
const double x = 1.0 - sin2_phi;
const double y = 1.0 - k*k*sin2_phi;
if(x < GSL_DBL_EPSILON) {
gsl_sf_result re;
const int status = gsl_sf_ellint_Ecomp_e(k, mode, &re);
/* could use A&S 17.4.14 to improve the value below */
result->val = 2*nc*re.val + GSL_SIGN(sin_phi) * re.val;
result->err = 2*fabs(nc)*re.err + re.err;
return status;
}
else {
gsl_sf_result rf, rd;
const double sin3_phi = sin2_phi * sin_phi;
const int rfstatus = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
const int rdstatus = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
result->val = sin_phi * rf.val - k*k/3.0 * sin3_phi * rd.val;
result->err = GSL_DBL_EPSILON * fabs(sin_phi * rf.val);
result->err += fabs(sin_phi*rf.err);
result->err += k*k/3.0 * GSL_DBL_EPSILON * fabs(sin3_phi * rd.val);
result->err += k*k/3.0 * fabs(sin3_phi*rd.err);
if (nc == 0) {
return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
} else {
gsl_sf_result re; /* add extra terms from periodicity */
const int restatus = gsl_sf_ellint_Ecomp_e(k, mode, &re);
result->val += 2*nc*re.val;
result->err += 2*fabs(nc)*re.err;
return GSL_ERROR_SELECT_3(rfstatus, rdstatus, restatus);
}
}
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.3)] */
int
gsl_sf_ellint_P_e(double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
{
/* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
exact reduction but this will have to do for now) BJG */
double nc = floor(phi/M_PI + 0.5);
double phi_red = phi - nc * M_PI;
phi = phi_red;
/* FIXME: need to handle the case of small x, as for E,F */
{
const double sin_phi = sin(phi);
const double sin2_phi = sin_phi * sin_phi;
const double sin3_phi = sin2_phi * sin_phi;
const double x = 1.0 - sin2_phi;
const double y = 1.0 - k*k*sin2_phi;
gsl_sf_result rf;
gsl_sf_result rj;
const int rfstatus = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
const int rjstatus = gsl_sf_ellint_RJ_e(x, y, 1.0, 1.0 + n*sin2_phi, mode, &rj);
result->val = sin_phi * rf.val - n/3.0*sin3_phi * rj.val;
result->err = GSL_DBL_EPSILON * fabs(sin_phi * rf.val);
result->err += fabs(sin_phi * rf.err);
result->err += n/3.0 * GSL_DBL_EPSILON * fabs(sin3_phi*rj.val);
result->err += n/3.0 * fabs(sin3_phi*rj.err);
if (nc == 0) {
return GSL_ERROR_SELECT_2(rfstatus, rjstatus);
} else {
gsl_sf_result rp; /* add extra terms from periodicity */
const int rpstatus = gsl_sf_ellint_Pcomp_e(k, n, mode, &rp);
result->val += 2*nc*rp.val;
result->err += 2*fabs(nc)*rp.err;
return GSL_ERROR_SELECT_3(rfstatus, rjstatus, rpstatus);
}
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.4)] */
int
gsl_sf_ellint_D_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
{
/* Angular reduction to -pi/2 < phi < pi/2 (we should really use an
exact reduction but this will have to do for now) BJG */
double nc = floor(phi/M_PI + 0.5);
double phi_red = phi - nc * M_PI;
phi = phi_red;
/* FIXME: need to handle the case of small x, as for E,F */
{
const double sin_phi = sin(phi);
const double sin2_phi = sin_phi * sin_phi;
const double sin3_phi = sin2_phi * sin_phi;
const double x = 1.0 - sin2_phi;
const double y = 1.0 - k*k*sin2_phi;
gsl_sf_result rd;
const int status = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
result->val = sin3_phi/3.0 * rd.val;
result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(sin3_phi/3.0 * rd.err);
if (nc == 0) {
return status;
} else {
gsl_sf_result rd; /* add extra terms from periodicity */
const int rdstatus = gsl_sf_ellint_Dcomp_e(k, mode, &rd);
result->val += 2*nc*rd.val;
result->err += 2*fabs(nc)*rd.err;
return GSL_ERROR_SELECT_2(status, rdstatus);
}
}
}
int
gsl_sf_ellint_Dcomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
{
if(k*k >= 1.0) {
DOMAIN_ERROR(result);
} else {
const double y = 1.0 - k*k; /* FIXME: still need to handle k~=~1 */
gsl_sf_result rd;
const int status = gsl_sf_ellint_RD_e(0.0, y, 1.0, mode, &rd);
result->val = (1.0/3.0) * rd.val;
result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(1.0/3.0 * rd.err);
return status;
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.5)] */
int
gsl_sf_ellint_Kcomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
{
if(k*k >= 1.0) {
DOMAIN_ERROR(result);
}
else if(k*k >= 1.0 - GSL_SQRT_DBL_EPSILON) {
/* [Abramowitz+Stegun, 17.3.34] */
const double y = 1.0 - k*k;
const double a[] = { 1.38629436112, 0.09666344259, 0.03590092383 };
const double b[] = { 0.5, 0.12498593597, 0.06880248576 };
const double ta = a[0] + y*(a[1] + y*a[2]);
const double tb = -log(y) * (b[0] + y*(b[1] + y*b[2]));
result->val = ta + tb;
result->err = 2.0 * GSL_DBL_EPSILON * (fabs(result->val) + fabs(k/y));
return GSL_SUCCESS;
}
else {
/* This was previously computed as,
return gsl_sf_ellint_RF_e(0.0, 1.0 - k*k, 1.0, mode, result);
but this underestimated the total error for small k, since the
argument y=1-k^2 is not exact (there is an absolute error of
GSL_DBL_EPSILON near y=0 due to cancellation in the subtraction).
Taking the singular behavior of -log(y) above gives an error
of 0.5*epsilon/y near y=0. (BJG) */
double y = 1.0 - k*k;
int status = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, result);
result->err += 0.5 * GSL_DBL_EPSILON / y;
return status ;
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.6)] */
int
gsl_sf_ellint_Ecomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
{
if(k*k >= 1.0) {
DOMAIN_ERROR(result);
}
else if(k*k >= 1.0 - GSL_SQRT_DBL_EPSILON) {
/* [Abramowitz+Stegun, 17.3.36] */
const double y = 1.0 - k*k;
const double a[] = { 0.44325141463, 0.06260601220, 0.04757383546 };
const double b[] = { 0.24998368310, 0.09200180037, 0.04069697526 };
const double ta = 1.0 + y*(a[0] + y*(a[1] + a[2]*y));
const double tb = -y*log(y) * (b[0] + y*(b[1] + b[2]*y));
result->val = ta + tb;
result->err = 2.0 * GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else {
gsl_sf_result rf;
gsl_sf_result rd;
const double y = 1.0 - k*k;
const int rfstatus = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, &rf);
const int rdstatus = gsl_sf_ellint_RD_e(0.0, y, 1.0, mode, &rd);
result->val = rf.val - k*k/3.0 * rd.val;
result->err = rf.err + k*k/3.0 * rd.err;
return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.3) phi=pi/2] */
int
gsl_sf_ellint_Pcomp_e(double k, double n, gsl_mode_t mode, gsl_sf_result * result)
{
if(k*k >= 1.0) {
DOMAIN_ERROR(result);
}
/* FIXME: need to handle k ~=~ 1 cancellations */
else {
gsl_sf_result rf;
gsl_sf_result rj;
const double y = 1.0 - k*k;
const int rfstatus = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, &rf);
const int rjstatus = gsl_sf_ellint_RJ_e(0.0, y, 1.0, 1.0 + n, mode, &rj);
result->val = rf.val - (n/3.0) * rj.val;
result->err = rf.err + fabs(n/3.0) * rj.err;
return GSL_ERROR_SELECT_2(rfstatus, rjstatus);
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_ellint_Kcomp(double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_Kcomp_e(k, mode, &result));
}
double gsl_sf_ellint_Ecomp(double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_Ecomp_e(k, mode, &result));
}
double gsl_sf_ellint_Pcomp(double k, double n, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_Pcomp_e(k, n, mode, &result));
}
double gsl_sf_ellint_Dcomp(double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_Dcomp_e(k, mode, &result));
}
double gsl_sf_ellint_F(double phi, double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_F_e(phi, k, mode, &result));
}
double gsl_sf_ellint_E(double phi, double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_E_e(phi, k, mode, &result));
}
double gsl_sf_ellint_P(double phi, double k, double n, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_P_e(phi, k, n, mode, &result));
}
double gsl_sf_ellint_D(double phi, double k, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_D_e(phi, k, mode, &result));
}
double gsl_sf_ellint_RC(double x, double y, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_RC_e(x, y, mode, &result));
}
double gsl_sf_ellint_RD(double x, double y, double z, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_RD_e(x, y, z, mode, &result));
}
double gsl_sf_ellint_RF(double x, double y, double z, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_RF_e(x, y, z, mode, &result));
}
double gsl_sf_ellint_RJ(double x, double y, double z, double p, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_RJ_e(x, y, z, p, mode, &result));
}